Lower bounds of Tian's invariant under toric invariances

On some toric Fano manifolds with metrics in the first Chern class, we show that a large family of smooth almost pluri-subharmonic functions (i.e. subharmonic with respect to the metric) with maximum equal to 0 admits a lower envelope. In our previous papers (A. Ben Abdesselem (2006) [4], and A. Ben Abdesselem and B. Dridi (2008) [5]) we established such envelopes when the functions considered are invariant under the action of a larger automorphisms group. Here we only consider the invariances due to the of the toric structure of the manifolds.